SpringerLink
Log in

Joint Continuity of Lyapunov Exponent for Finitely Smooth Quasi-periodic Schrödinger Cocycles

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We prove the joint continuity of Lyapunov exponent in the energy and the Diophantine frequency for quasi-periodic Schrödinger cocycles with the C2 cos-type potentials. In particular, the Lyapunov exponent is log-Hölder continuous at each Diophantine frequency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avila, A., Jitomirskaya, S., Sadel, C.: Complex one-frequency cocycles. J. Eur. Math. Soc., 16, 1915–1935 (2013)

    Article  MathSciNet  Google Scholar 

  2. Bochi, J.: Discontinuity of the Lyapunov exponent for non-hyperbolic cocycles, unpublished, 1999 https://www.researchgate.net/publication/262012858

  3. Bochi, J.: Genericity of zero Lyapunov exponents. Ergodic Theory Dynam. Systems, 22, 1667–1696 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Bourgain, J.: Positivity and continuity of the Lyapunov exponent for shifts on \({{\mathbb{T}}^d}\) with arbitrary frequency vector and real analytic potential. J. Anal. Math., 96, 313–355 (2005)

    Article  MathSciNet  Google Scholar 

  5. Bourgian, J., Goldstein, M.: On nonperturbative localization with quasi-periodic potential. Ann. of Math., 152, 835–879 (2000)

    Article  MathSciNet  Google Scholar 

  6. Bourgain, J., Jitomirskaya, S.: Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J. Stat. Phys., 108, 1203–1218 (2000)

    Article  MathSciNet  Google Scholar 

  7. Cai, A., Chavaudret, C., You, J., et al.: Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles. Math. Z., 291, 931–968 (2019)

    Article  MathSciNet  Google Scholar 

  8. Furman, A.: On the multiplicative ergodic theorem for the uniquely ergodic systems. Ann. Inst. H. Poincaré Probab. Statist., 33, 797–815 (1997)

    Article  MathSciNet  Google Scholar 

  9. Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of Math., 154, 155–203 (2001)

    Article  MathSciNet  Google Scholar 

  10. Jitomirskaya, S., Koslover, D., Schulteis, M.: Continuity of the Lyapunov exponent for general analytic quasiperiodic cocycles. Ergodic Theory Dynam. Systems, 29, 1881–1905 (2009)

    Article  MathSciNet  Google Scholar 

  11. Jitomirskaya, S., Marx, C.: Continuity of the Lyapunov exponent for analytic quasi-periodic cocycles with singularities. J. Fixed Point Theory Appl., 10, 129–146 (2011)

    Article  MathSciNet  Google Scholar 

  12. Jitomirskaya, S., Marx, C.: Analytic quasi-periodic cocycles with singularities and the Lyapunov exponent of extended Harpers model. Comm. Math. Phys., 316, 237–267 (2012)

    Article  MathSciNet  Google Scholar 

  13. Klein, S.: Anderson localization for the discrete one-dimensional quasi-periodic Schroödinger operator with potential defined by a Gevrey-class function. J. Funct. Anal., 218, 255–292 (2005)

    Article  MathSciNet  Google Scholar 

  14. Knill, O.: The upper Lyapunov exponent of SL(2, ℝ) cocycles: Discontinuity and the problem of positivity. Lecture Notes in Math., 1486, Lyapunov Exponents (Oberwolfach, 1990) 86–97 (1991)

  15. Liang, J., Kung, P.: Positivity of Lyapunov exponent for a class of smooth Schroödinger cocycles with weak Liouville frequencies. Front. Math. China, 12, 607–639 (2017)

    Article  MathSciNet  Google Scholar 

  16. Liang, J., Wang, Y., You, J.: Höolder continuity of Lyapunov exponent for a family of smooth Schröodinger cocycles, preprint (2018) https://arxiv.org/pdf/1806.03284.pdf

  17. Thouvenot, J.: An example of discontinuity in the computation of the Lyapunov exponents. Proc. Steklov Inst. Math., 216, 366–369 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Wang, Y., You, J.: Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles. Duke Math. J., 162, 2363–2412 (2013)

    Article  MathSciNet  Google Scholar 

  19. Wang, Y., You, J.: Examples of non-openness of smooth quasiperiodic cocycles with positive Lyapunov exponent. Comm. Math. Phys., 362, 801–826 (2018)

    Article  MathSciNet  Google Scholar 

  20. Wang, Y., Zhang, Z.: Uniform positivity and continuity of Lyapunov exponents for a class of C2 quasiperiodic Schrödinger cocycles. J. Funct. Anal., 268, 2525–2585 (2015)

    Article  MathSciNet  Google Scholar 

  21. Xu, J., Ge, L., Wang, Y.: The Hölder continuity of Lyapunov exponents for a class of cos-type quasiperiodic Schrödinger cocycles, preprint (2019) https://arxiv.org/pdf/2006.03381.pdf

  22. Young, L.: Lyapunov exponents for some quasi-periodic cocycles. Ergodic Theory Dynam. Systems, 17, 483–504 (1997)

    Article  MathSciNet  Google Scholar 

  23. Zhang, Z.: Positive Lyapunov exponents for quasiperiodic Szegö cocycles. Nonlinearity, 25, 1771–1797 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Nan**g Forestry University, Nan**g, 210037, P. R. China

    ** Hao Liang

  2. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China

    Lin Lin Fu

Authors
  1. ** Hao Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lin Lin Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lin Lin Fu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liang, J.H., Fu, L.L. Joint Continuity of Lyapunov Exponent for Finitely Smooth Quasi-periodic Schrödinger Cocycles. Acta. Math. Sin.-English Ser. 37, 1131–1142 (2021). https://doi.org/10.1007/s10114-021-9137-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-021-9137-y

Keywords

MR(2010) Subject Classification

Search

Navigation